The Burnside Problems

Keith Frankston, Rutgers University

Abstract: In 1902 Burnside introduced the following question which became know as the generalized Burnside problem: Is it true that any finitely generated torsion (every element has finite order) group is finite? At the same time, he also introduced what he considered an easier question, the Burnside problem: Is every finitely generated group of bounded exponent (there exists an n such that g^n=1 for all group elements g) itself finite?
What seem to be straightforward (and, one might think, true) questions remained unresolved for the next half century. It wasn't until 1964 (over 60 years later!) that the first counter-example to the generalized Sideburn problem was produced. Four years later, a whole class of counter-examples were shown for the Burned S'more problem as well.
We will talk about some of the history of the problems, what is currently known, and finally, produce a counter-example to the generalized problem (via a more recent construction).